Vector bundles on quantum conjugacy classes

TL;DR

This paper constructs a framework for quantizing equivariant vector bundles on conjugacy classes of classical Lie groups using categories of modules over quantum groups, revealing their semi-simple structure.

Contribution

It introduces categories of modules associated with points on the maximal torus and explicitly quantizes equivariant vector bundles on conjugacy classes.

Findings

Categories are essentially semi-simple.

Explicit quantization of equivariant vector bundles.

Framework applies to classical Lie groups.

Abstract

Let $\mathfrak{g}$ be a simple complex Lie algebra of a classical type and $U_q(\mathfrak{g})$ the corresponding Drinfeld-Jimbo quantum group at $q$ not a root of unity. With every point $t$ of the fixed maximal torus $T$ of an algebraic group $G$ with Lie algebra $\mathfrak{g}$ we associate an additive category $\mathcal{O}_q(t)$ of $U_q(\mathfrak{g})$-modules that is stable under tensor product with finite-dimensional quasi-classical $U_q(\mathfrak{g})$-modules. We prove that $\mathcal{O}_q(t)$ is essentially semi-simple and use it to explicitly quantize equivariant vector bundles on the conjugacy class of $t$.